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A new approach to light bulb tricks: Disks in 4-manifolds

Danica Kosanović, Peter Teichner

TL;DR

The paper extends Gabai’s light bulb theorem to disks in 4-manifolds with boundary by introducing the relative Dax invariant $\mathbb{D}\mathsf{ax}$, which, together with homotopy data, classifies neat disks with fixed boundary. It reveals a natural geometric action of $\mathbb{Z}[\pi\setminus 1]$ on the isotopy classes $\mathbb{D}(M;k)$, yielding a 2-step nilpotent group structure and explicit commutator formulas tied to the reduced intersection form $\lambda$ and Wall invariants $\mu_2,\mu_3$. The framework unifies and extends prior sphere results (Gabai, ST-LBT) by connecting disk theory to Freedman–Quinn invariants and deriving applications to mapping class groups via ambient isotopies across dual spheres. The approach, grounded in space-level Cerf/Dax methods and KT-highd, provides concrete realizations of all possible Dax values, including highly nonabelian examples, and recovers sphere results in boundary-sphere dual scenarios with a boundary component diffeomorphic to $\mathbb{S}^1\times\mathbb{S}^2$. The results yield a robust toolkit for understanding embeddings, isotopy classes, and diffeomorphisms in 4-manifolds, with explicit constructions in boundary-connected-sum and boundary-dual configurations.

Abstract

For a 4-manifold $M$ and a knot $k\colon\mathbb{S}^1\hookrightarrow\partial M$ with dual sphere $G\colon\mathbb{S}^2\hookrightarrow\partial M$, we compute the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings $\mathbb{D}^2\hookrightarrow M$ with boundary $k$, using an invariant going back to Dax. Moreover, we construct a group structure on $\mathbb{D}(M;k)$ and show that it is usually neither abelian nor finitely generated. We recover all previous results for isotopy classes of spheres with framed duals and relate the group $\mathbb{D}(M;k)$ to the mapping class group of $M$.

A new approach to light bulb tricks: Disks in 4-manifolds

TL;DR

The paper extends Gabai’s light bulb theorem to disks in 4-manifolds with boundary by introducing the relative Dax invariant , which, together with homotopy data, classifies neat disks with fixed boundary. It reveals a natural geometric action of on the isotopy classes , yielding a 2-step nilpotent group structure and explicit commutator formulas tied to the reduced intersection form and Wall invariants . The framework unifies and extends prior sphere results (Gabai, ST-LBT) by connecting disk theory to Freedman–Quinn invariants and deriving applications to mapping class groups via ambient isotopies across dual spheres. The approach, grounded in space-level Cerf/Dax methods and KT-highd, provides concrete realizations of all possible Dax values, including highly nonabelian examples, and recovers sphere results in boundary-sphere dual scenarios with a boundary component diffeomorphic to . The results yield a robust toolkit for understanding embeddings, isotopy classes, and diffeomorphisms in 4-manifolds, with explicit constructions in boundary-connected-sum and boundary-dual configurations.

Abstract

For a 4-manifold and a knot with dual sphere , we compute the set of smooth isotopy classes of neat embeddings with boundary , using an invariant going back to Dax. Moreover, we construct a group structure on and show that it is usually neither abelian nor finitely generated. We recover all previous results for isotopy classes of spheres with framed duals and relate the group to the mapping class group of .
Paper Structure (22 sections, 32 theorems, 54 equations, 10 figures)

This paper contains 22 sections, 32 theorems, 54 equations, 10 figures.

Table of Contents

  1. Introduction and main results
  2. The relative Dax invariant for neat disks
  3. Classification of neat disks via a geometric group action
  4. Nilpotent group structure on neat disks
  5. Further results and relations to other work
  6. Application to mapping class groups of 4-manifolds
  7. Deriving the light bulb trick for spheres
  8. Key inputs from spaces of embeddings
  9. Spaces of embedded and immersed arcs
  10. The Dax invariant for spaces of arcs
  11. Group commutators and Whitehead products
  12. Some algebraic topology of the dax homomorphisms
  13. The relative Dax invariant for disks in 4-manifolds
  14. Dax invariant for half-disks and neat disks
  15. Dax invariant and Wall's self-intersection invariant
  16. ...and 7 more sections

Key Result

Theorem 1.1

For a knot $k\colon\mathbb{S}^1\hookrightarrow\partial M$ that has a dual sphere in $\partial M$, two neat embeddings $\mathbb{D}^2\hookrightarrow M$ with boundary $k$ are isotopic (rel. $k$) if and only if they are homotopic (rel. $k$) and their relative $\mathbb{D}\mathsf{ax}$ invariant vanishes.

Figures (10)

  • Figure 1.1: Constructing isotopy classes of disks with fixed boundary $k$ by finger moves along $g$ and Norman tricks on distinct sheets.
  • Figure 1.2: A handle diagram for $M_c$ with $\mu_2(S_c)=-t+t^2$.
  • Figure 1.3: The interior connected sum $K \# rG$, the immersed disk $K_{tw}$ obtained by an interior twist on $K$, and the resolved disk $K_{tw}^G$.
  • Figure 3.1: The double point $y_i\in X$ of the arc $F(\vec{t}_i)\in\mathop{\mathrm{Imm}}\nolimits_\partial(\mathbb{D}^1,X)$ has the associated loop $g_{y_i}=g^{-1}$.
  • Figure 3.2: The immersed arc in the family for $g\cdot[C]\in\pi_3M$.
  • ...and 5 more figures

Theorems & Definitions (78)

  • Theorem 1.1
  • Example 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Definition 1.6
  • Proposition 1.7
  • Lemma 1.9
  • Theorem 1.11
  • Remark 1.12
  • Example 1.13
  • ...and 68 more